Abstract
In this paper we consider a periodic boundary value problem for the generalized Ginzburg-Landau. The generalized version of the weakly dissipative Ginzburg-Landau equation differs from the traditional version by replacing the cubic nonlinearity with nonlinearity of arbitrary odd degree. We will show that the periodic boundary value problem has a countable set of solutions that are single-mode and periodic in the evolutionary variable. We will examine the stability question as well as local bifurcations of such solutions when they change stability. In this case, the two-dimensional attracting invariant tori bifurcate emerges when stability is lost from single-mode solutions. These are non-resonant tori that have appeared in the generic situation. The main results are obtained on the basis and development of methods of the theory of dynamical systems with an infinite-dimensional phase space. These include the method of invariant manifolds and normal forms, as well as the principle of self-similarity. This principle allows us to reduce the problem of bifurcations of a countable set of single-mode solutions to the analysis of the corresponding problem.
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Kulikov, A., Kulikov, D. (2023). Local Bifurcations of Periodic Traveling Waves in the Generalized Weakly Dissipative Ginzburg-Landau Equation. In: Vasilyev, V. (eds) Differential Equations, Mathematical Modeling and Computational Algorithms. DEMMCA 2021. Springer Proceedings in Mathematics & Statistics, vol 423. Springer, Cham. https://doi.org/10.1007/978-3-031-28505-9_3
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